Recasting the Hazrat Conjecture: Relating Shift Equivalence to Graded Morita Equivalence

Let $E$ and $F$ be finite graphs with no sinks, and $k$ any field. We show that shift equivalence of the adjacency matrices $A_E$ and $A_F$, together with an additional compatibility condition, implies that the Leavitt path algebras $L_k(E)$ and $L_k(F)$ are graded Morita equivalent. Along the way, we build a new type of $L_k(E)$–$L_k(F)$-bimodule (a bridging bimodule), which we use to establish the graded equivalence.